Optimal relations between Lp-norms for the Hardy operator and its dual

TL;DR

This paper establishes precise two-sided inequalities relating the L^p norms of the Hardy operator and its dual, providing sharp constants and extending previous results to a broader range of p values.

Contribution

It offers a unified, sharp characterization of L^p norm relations for the Hardy operator and its dual, improving and generalizing earlier findings.

Findings

Derived sharp two-sided inequalities for L^p norms of Hf and H*f.

Extended known results to the full range 1<p<2 with sharp constants.

Provided alternative proofs for existing inequalities in special cases.

Abstract

We obtain sharp two-sided inequalities between $L^p-$norms $(1<p<\infty)$ of functions $Hf$ and $H^*f$, where $H$ is the Hardy operator, $H^*$ is its dual, and $f$ is a nonnegative measurable function on $(0,\infty).$ In an equivalent form, it gives sharp constants in the two-sided relations between $L^p$-norms of functions $H\f-\f$ and $\f$, where $\f$ is a nonnegative nonincreasing function on $(0,+\infty)$ with $\f(+\infty)=0.$ In particular, it provides an alternative proof of a result obtained by N. Kruglyak and E. Setterqvist (2008) for $p=2k (k\in \N)$ and by S. Boza and J. Soria (2011) for all $p\ge 2$, and gives a sharp version of this result for $1<p<2$.